Are the ring $\mathbb{R}[x]/(x^{2})$ and $\mathbb{R}[x]/(x^{3})$ isomorphic?

Question

Are the ring $\mathbb{R}[x]/(x^{2})$ and $\mathbb{R}[x]/(x^{3})$ isomorphic? Justify your answer

I have no idea to do this question, can anyone help me?

Answer 1

Is there an element $y\in\mathbb{R}[x]/(x^2)$, like $x$ in $\mathbb{R}[x]/(x^3)$, such that $y^2\neq 0$ but $y^3=0$? Well, $$y=a+bx$$ for some $a,b\in\mathbb{R}$. Necessarily $a\neq 0$ and $b\neq 0$. $$y^2=a^2+2abx$$ and $$y^3=a^3+3a^2bx$$ Neither $y^2$ nor $y^3$ is $0$, so there is no such element in $\mathbb{R}[x]/(x^2)$. Hence the two rings cannot be isomorphic.

Answer 2

There's always something for you to try to do.

For example, what do the elements of $\mathbb R [x]/(x^2)$ and $\mathbb R [x]/(x^3)$ look like? If you can answer this, then it problem becomes much clearer.

Hint: To find elements of $\mathbb R [x]/(q(x))$: for any $f(x) \in \mathbb R[x]$, do polynomial division with $q(x)$, so that we may write $f(x) = g(x) q(x) + r(x)$, for some $r(x)$ with degree less than $q(x)$...